(Note: This is related to the post on Laplace’s rule).
For each interval length \(s\) we have a sequence of binary variables \(Y_{1}(s),Y_{2}(s),\ldots Y_{n/s}(s)\). We want to connect these variables, for each \(s\), in a meaningful way and apply Laplace’s rule. One way to do so is by assuming a latent Poisson process.
Let \(X_{t}\) be a Poisson process with parameter \(\lambda\), \(t\in[0,n]\). Then the number of observations in the time frame \([t,t+s]\) is distributed as $$ X_{t+s}-X_{t}\sim\text{Poisson}(\lambda s). $$ A natural way to make \(X_{t+s}-X_{t}\) into a binary variable is to use $$ Y_{i}(s)=1[X_{t+s}-X_{t}=0]. $$ Then it follows that $$ P(Y_{i}(s)=1)=e^{-\lambda s}. $$ To apply Laplace’s rule on a fixed time scale \(s_{0}\), such as days, we’ll have to make the prior over \(\pi(s_{0})=e^{-\lambda s_{0}}\) uniform. This is equivalent to \(\lambda s_{0}\sim\textrm{Exp}(1)\), or \(\lambda\sim\textrm{Exp}(1/s_{0})\).
Now, if we want to look at another time scale \(s_{1}\), the induced prior for the success propability \(\pi(s_{1})=e^{-\lambda s_{1}}\) is $$ \pi(s_{1})=\pi(s_{0}){}^{\frac{s_{1}}{s_{0}}}\sim\text{Beta}(s_{0}/s_{1},1). $$